27. Dark energy 1

27. Dark Energy

Revised September 2019 by D. H. Weinberg (OSU) and M. White (UCB, LBL)

27.1. Repulsive Gravity and Cosmic Acceleration

In the first modern cosmological model, Einstein [1] modified his field equation ofGeneral Relativity (GR), introducing a “cosmological term” that enabled a solution withtime-independent, spatially homogeneous matter density ρm and constant positive spacecurvature. Although Einstein did not frame it this way, one can view the “cosmologicalconstant” Λ as representing a constant energy density of the vacuum [2], whose repulsivegravitational effect balances the attractive gravity of matter and thereby allows a staticsolution. After the development of dynamic cosmological models [3,4] and the discoveryof cosmic expansion [5], the cosmological term appeared unnecessary, and Einstein andde Sitter [6] advocated adopting an expanding, homogeneous and isotropic, spatiallyflat, matter-dominated Universe as the default cosmology until observations dictatedotherwise. Such a model has matter density equal to the critical density, Ωm ≡ ρm/ρc = 1,and negligible contribution from other energy components [7].

By the mid-1990s, the Einstein-de Sitter model was showing numerous cracks,under the combined onslaught of data from the cosmic microwave background (CMB),large-scale galaxy clustering, and direct estimates of the matter density, the expansionrate (H0), and the age of the Universe. As noted in a number of papers from thistime, introducing a cosmological constant offered a potential resolution of many of thesetensions, yielding the most empirically successful version of the inflationary cold darkmatter scenario. In the late 1990s, supernova surveys by two independent teams provideddirect evidence for accelerating cosmic expansion [8,9], establishing the cosmologicalconstant model (with Ωm ≃ 0.3, ΩΛ ≃ 0.7) as the preferred alternative to the Ωm = 1scenario. Shortly thereafter, CMB evidence for a spatially flat Universe [10,11], andthus for Ωtot ≃ 1, cemented the case for cosmic acceleration by firmly eliminating thefree-expansion alternative with Ωm ≪ 1 and ΩΛ = 0. Today, the accelerating Universeis well established by multiple lines of independent evidence from a tight web of precisecosmological measurements.

As discussed in the Big Bang Cosmology article of this Review (Sec. 21), the scalefactor R(t) of a homogeneous and isotropic Universe governed by GR grows at anaccelerating rate if the pressure p < −1

3ρ (in c = 1 units). A cosmological constant hasρΛ = constant and pressure pΛ = −ρΛ (see Eq. 21.10), so it will drive acceleration if itdominates the total energy density. However, acceleration could arise from a more generalform of “dark energy” that has negative pressure, typically specified in terms of theequation-of-state-parameter w = p/ρ (= −1 for a cosmological constant). Furthermore,the conclusion that acceleration requires a new energy component beyond matter andradiation relies on the assumption that GR is the correct description of gravity oncosmological scales. The title of this article follows the common but inexact usage of“dark energy” as a catch-all term for the origin of cosmic acceleration, regardless ofwhether it arises from a new form of energy or a modification of GR. Our account heredraws on the much longer review of cosmic acceleration by Ref. [12], which providesbackground explanation and extensive literature references for the discussion in Secs.27.2 and 27.3.

M. Tanabashi et al. (Particle Data Group), Phys. Rev. D 98, 030001 (2018) and 2019 updateDecember 6, 2019 12:03

2 27. Dark energy

Below we will use the abbreviation ΛCDM to refer to a model with cold dark matter,a cosmological constant, inflationary initial conditions, standard radiation and neutrinocontent, and a flat Universe with Ωtot = 1 (though we will sometimes describe this modelas “flat ΛCDM” to emphasize this last restriction). We will use wCDM to denote a modelwith the same assumptions but a free, constant value of w. Models with the prefix “o”(e.g., owCDM) allow non-zero space curvature.

27.2. Theories of Cosmic Acceleration

27.2.1. Dark Energy or Modified Gravity? :

A cosmological constant is the mathematically simplest, and perhaps the physicallysimplest, theoretical explanation for the accelerating Universe. The problem is explainingits unnaturally small magnitude, as discussed in Sec. 21.4.7 of this Review. An alternative(which still requires finding a way to make the cosmological constant zero or at leastnegligibly small) is that the accelerating cosmic expansion is driven by a new form ofenergy such as a scalar field [13] with potential V (φ). The energy density and pressure ofthe field φ(x) take the same forms as for inflationary scalar fields, given in Eq. (21.52) ofthe Big Bang Cosmology article. In the limit that 1

2 φ2 ≪ |V (φ)|, the scalar field acts like

a cosmological constant, with pφ ≃ −ρφ. In this scenario, today’s cosmic acceleration isclosely akin to the epoch of inflation, but with radically different energy and timescale.

More generally, the value of w = pφ/ρφ in scalar field models evolves with time in a

way that depends on V (φ) and on the initial conditions (φi, φi); some forms of V (φ) haveattractor solutions in which the late-time behavior is insensitive to initial values. Manyforms of time evolution are possible, including ones where w is approximately constantand broad classes where w “freezes” towards or “thaws” away from w = −1, with thetransition occurring when the field comes to dominate the total energy budget. If ρφ iseven approximately constant, then it becomes dynamically insignificant at high redshift,because the matter density scales as ρm ∝ (1 + z)3. “Early dark energy” models are onesin which ρφ is a small but not negligible fraction (e.g., a few percent) of the total energythroughout the matter- and radiation-dominated eras, tracking the dominant componentbefore itself coming to dominate at low redshift.

Instead of introducing a new energy component, one can attempt to modify gravityin a way that leads to accelerated expansion [14]. One option is to replace the Ricciscalar R with a function R + f(R) in the gravitational action [15]. Other changes canbe more radical, such as introducing extra dimensions and allowing gravitons to “leak”off the brane that represents the observable Universe (the “DGP” model [16]) . TheDGP example has inspired a more general class of “galileon” and massive gravity models.Constructing viable modified gravity models is challenging, in part because it is easy tointroduce theoretical inconsistencies (such as “ghost” fields with negative kinetic energy),but above all because GR is a theory with many high-precision empirical successes onsolar system scales [17]. Modified gravity models typically invoke screening mechanismsthat force model predictions to approach those of GR in regions of high density or stronggravitational potential. Screening offers potentially distinctive signatures, as the strength

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27. Dark energy 3

of gravity (i.e., the effective value of GN) can vary by order unity in environments withdifferent gravitational potentials.

More generally, one can search for signatures of modified gravity by comparing thehistory of cosmic structure growth to the history of cosmic expansion. Within GR, thesetwo are linked by a consistency relation, as described below (Eq. (27.2)). Modifyinggravity can change the predicted rate of structure growth, and it can make the growthrate dependent on scale or environment. In some circumstances, modifying gravity altersthe combinations of potentials responsible for gravitational lensing and the dynamics ofnon-relativistic tracers (such as galaxies or stars) in different ways (see Sec. 21.4.7 in thisReview), leading to order unity mismatches between the masses of objects inferred fromlensing and those inferred from dynamics in unscreened environments.

At present there are no fully realized and empirically viable modified gravity theoriesthat explain the observed level of cosmic acceleration. The constraints on f(R) modelsnow force them so close to GR that they cannot produce acceleration without introducinga separate dark energy component [18]. The DGP model is empirically ruled outby several tests, including the expansion history, the integrated Sachs-Wolfe effect,and redshift-space distortion measurements of the structure growth rate [19]. Thenear-simultaneous arrival of gravitational waves and electromagnetic signals from theneutron star merger event GW170817, which shows that gravitational waves travel atalmost exactly the speed of light, is a further strong constraint on modified gravitytheories [20]. The elimination of models should be considered an important success ofthe program to empirically test theories of cosmic acceleration. However, it is worthrecalling that there was no fully realized gravitational explanation for the precession ofMercury’s orbit prior to the completion of GR in 1915, and the fact that no completeand viable modified gravity theory exists today does not mean that one will not arise inthe future. In the meantime, we can continue empirical investigations that can tightenrestrictions on such theories or perhaps point towards the gravitational sector as theorigin of accelerating expansion.

27.2.2. Expansion History and Growth of Structure :

The main line of empirical attack on dark energy is to measure the history of cosmicexpansion and the history of matter clustering with the greatest achievable precisionover a wide range of redshift. Within GR, the expansion rate H(z) is governed bythe Friedmann equation (see the articles on Big Bang Cosmology and CosmologicalParameters—Secs. 21 and 24 in this Review). For dark energy with an equation of statew(z), the cosmological constant contribution to the expansion, ΩΛ, is replaced by aredshift-dependent contribution. The evolution of the dark energy density follows fromEq. (21.10),

Ωdeρde(z)

ρde(z = 0)= Ωde exp

[

3

∫ z

0[1 + w(z′)]

dz′

1 + z′

]

= Ωde(1 + z)3(1+w), (27.1)

where the second equality holds for constant w. If Ωm, Ωr, and the present value of Ωtot

are known, then measuring H(z) pins down w(z). (Note that Ωde is the same quantity

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4 27. Dark energy

denoted Ωv in Sec. 21, but we have adopted the ‘de’ subscript to avoid implying thatdark energy is necessarily a vacuum effect.)

While some observations can probe H(z) directly, others measure the distance-redshiftrelation. The basic relations between angular diameter distance or luminosity distanceand H(z) are given in Ch. 21 —and these are generally unaltered in time-dependent darkenergy or modified gravity models. For convenience, in later sections, we will sometimesrefer to the comoving angular distance, DA,c(z) = (1 + z)DA(z).

In GR-based linear perturbation theory, the density contrast δ(x, t) ≡ ρ(x, t)/ρ(t) − 1of pressureless matter grows in proportion to the linear growth function G(t) (not to beconfused with the gravitational constant GN), which follows the differential equation

G+ 2H(z)G−3

2ΩmH2

0 (1 + z)3G = 0 . (27.2)

To a good approximation, the logarithmic derivative of G(z) is

f(z) ≡ −d lnG

d ln(1 + z)≃

[

Ωm(1 + z)3H2

0

H2(z)

]γ

, (27.3)

where γ ≃ 0.55 for relevant values of cosmological parameters [21]. In an Ωm = 1Universe, G(z) ∝ (1 + z)−1, but growth slows when Ωm drops significantly below unity.One can integrate Eq. (27.3) to get an approximate integral relation between G(z)and H(z), but the full (numerical) solution to Eq. (27.2) should be used for precisioncalculations. Even in the non-linear regime, the amplitude of clustering is determinedmainly by G(z), so observations of non-linear structure can be used to infer the linearG(z), provided one has good theoretical modeling to relate the two.

In modified gravity models the growth rate of gravitational clustering may differ fromthe GR prediction. A general strategy to test modified gravity, therefore, is to measureboth the expansion history and the growth history to see whether they yield consistentresults for H(z) or w(z).

27.2.3. Parameters :

Constraining a general history of w(z) is nearly impossible, because the dark energydensity, which affects H(z), is given by an integral over w(z), and distances and thegrowth factor involve a further integration over functions of H(z). Oscillations in w(z)over a range ∆z/(1 + z) ≪ 1 are therefore extremely difficult to constrain. It has becomeconventional to phrase constraints or projected constraints on w(z) in terms of a linearevolution model,

w(a) = w0 + wa(1− a) = wp + wa(ap − a), (27.4)

where a ≡ (1 + z)−1, w0 is the value of w at z = 0, and wp is the value of w at a “pivot”redshift zp ≡ a−1

p − 1, where it is best constrained by a given set of experiments. Fortypical data combinations, zp ≃ 0.5. This simple parameterization can provide a goodapproximation to the predictions of many physically motivated models for observablesmeasured with percent-level precision. A widely used “Figure of Merit” (FoM) for

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27. Dark energy 5

dark energy experiments [22] is the projected combination of errors [σ(wp)σ(wa)]−1.

Ambitious future experiments with 0.1–0.3% precision on observables can constrain richerdescriptions of w(z), which can be characterized by principal components.

There has been less convergence on a standard parameterization for describing modifiedgravity theories. Deviations from the GR-predicted growth rate can be described bya deviation ∆γ in the index of Eq. (27.3), together with an overall multiplicativeoffset relative to the G(z) expected from extrapolating the CMB-measured fluctuationamplitude to low redshift. However, these two parameters may not accurately capturethe growth predictions of all physically interesting models. Another important parameterto constrain is the ratio of the gravitational potentials governing space curvature and theacceleration of non-relativistic test particles. The possible phenomenology of modifiedgravity models is rich, which enables many consistency tests but complicates the task ofconstructing parameterized descriptions.

The more general set of cosmological parameters is discussed elsewhere in this Review(Sec. 24), but here we highlight a few that are particularly important to the dark energydiscussion.

• The dimensionless Hubble parameter h ≡ H0/100 km s−1Mpc−1 determines thepresent day value of the critical density and the overall scaling of distances inferredfrom redshifts.

• Ωm and Ωtot affect the expansion history and the distance-redshift relation.

• The sound horizon rs =∫ trec0 cs(t)dt/a(t), the comoving distance that pressure waves

can propagate between t = 0 and recombination, determines the physical scale ofthe acoustic peaks in the CMB and the baryon acoustic oscillation (BAO) feature inlow-redshift matter clustering [23].

• The amplitude of matter fluctuations, conventionally represented by the quantityσ8(z), scales the overall amplitude of growth measures such as weak lensing orredshift-space distortions (discussed in the next section).

Specifically, σ8(z) refers to the rms fluctuation of the matter overdensity ρ/ρ in spheresof radius 8h−1Mpc, computed from the linear theory matter power spectrum at redshiftz, and σ8 on its own refers to the value at z = 0 (just like our convention for Ωm).

While discussions of dark energy are frequently phrased in terms of values and errorson quantities like wp, wa, ∆γ, and Ωtot, parameter precision is the means to an end,not an end in itself. The underlying goal of empirical studies of cosmic acceleration is toaddress two physically profound questions:

1. Does acceleration arise from a breakdown of GR on cosmological scales or from anew energy component that exerts repulsive gravity within GR?

2. If acceleration is caused by a new energy component, is its energy density constantin space and time, as expected for a fundamental vacuum energy, or does it showvariations that indicate a dynamical field?

Substantial progress towards answering these questions, in particular any definitiverejection of the cosmological constant “null hypothesis,” would be a major breakthroughin cosmology and fundamental physics.

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6 27. Dark energy

27.3. Observational Probes

We briefly summarize the observational probes that play the greatest role in currentconstraints on dark energy. Further discussion can be found in other articles of thisReview, in particular Secs. 24 (Cosmological Parameters) and 28 (The Cosmic MicrowaveBackground), and in Ref. [12], which provides extensive references to backgroundliterature. Recent observational results from these methods are discussed in Sec. 27.4.

27.3.1. Methods, Sensitivity, Systematics :

Cosmic Microwave Background Anisotropies: Although CMB anisotropies provide limitedinformation about dark energy on their own, CMB constraints on the geometry, mattercontent, and radiation content of the Universe play a critical role in dark energy studieswhen combined with low-redshift probes. In particular, CMB data supply measurementsof θs = rs/DA,c(zrec), the angular size of the sound horizon at recombination, from

the angular location of the acoustic peaks, measurements of Ωmh2 and Ωbh2 from the

heights of the peaks, and normalization of the amplitude of matter fluctuations at zrecfrom the amplitude of the CMB fluctuations themselves. Planck data yield a 0.18%determination of rs, which scales as (Ωmh2)−0.25 for cosmologies with standard matterand radiation content. The uncertainty in the matter fluctuation amplitude at the epochof recombination is 0.5%. Secondary anisotropies, including the integrated Sachs-Wolfeeffect, the Sunyaev-Zeldovich (SZ, [24]) effect, and weak lensing of primary anisostropies,provide additional information about dark energy by constraining low-redshift structuregrowth.

Type Ia Supernovae (SN): Type Ia supernovae, produced by the thermonuclear explosionsof white dwarfs, exhibit 10–15% scatter in peak luminosity after correction for light curveduration (the time to rise and fall) and color (which is a diagnostic of dust extinction).Since the peak luminosity is not known a priori, supernova surveys constrain ratios ofluminosity distances at different redshifts. If one is comparing a high-redshift sample toa local calibrator sample measured with much higher precision (and distances inferredfrom Hubble’s law), then one essentially measures the luminosity distance in h−1Mpc,constraining the combination hDL(z). With distance uncertainties of 5–8% per wellobserved supernova, a sample of around 100 SNe is sufficient to achieve sub-percentstatistical precision. The 1–2% systematic uncertainties in current samples are dominatedby uncertainties associated with photometric calibration and dust extinction correctionsplus the observed dependence of luminosity on host galaxy properties. Another potentialsystematic is redshift evolution of the supernova population itself, which can be tested byanalyzing subsamples grouped by spectral properties or host galaxy properties to confirmthat they yield consistent results.

Baryon Acoustic Oscillations (BAO): Pressure waves that propagate in the pre-recombination photon-baryon fluid imprint a characteristic scale in the clustering ofmatter and galaxies, which appears in the galaxy correlation function as a localized peakat the sound horizon scale rs, or in the power spectrum as a series of oscillations. Sinceobserved galaxy coordinates consist of angles and redshifts, measuring this “standardruler” scale in a galaxy redshift survey determines the angular diameter distanceDA(z) and the expansion rate H(z), which convert coordinate separations to comoving

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27. Dark energy 7

distances. Errors on the two quantities are correlated, and in existing galaxy surveysthe best determined combination is approximately DV(z) = [czD2

A,c(z)/H(z)]1/3. As anapproximate rule of thumb, a survey that fully samples structures at redshift z overa comoving volume V , and is therefore limited by cosmic variance rather than shotnoise, measures DA,c(z) with a fractional error of 0.005(V/10Gpc3)−1/2 and H(z) witha fractional error 1.6–1.8 times higher. The most precise BAO measurements to datecome from large galaxy redshift surveys probing z < 0.8, and these will be extended tohigher redshifts by future projects. At redshifts z > 2, BAO can also be measured in theLyman-α forest of intergalactic hydrogen absorption towards background quasars, wherethe fluctuating absorption pattern provides tens or hundreds of samples of the densityfield along each quasar sightline. For Lyman-α forest BAO, the best measured parametercombination is more heavily weighted towards H(z) because of strong redshift-spacedistortions that enhance clustering in the line-of-sight direction. Radio intensity mapping,which maps large-scale structure in redshifted 21-cm hydrogen emission without resolvingindividual galaxies, offers a potentially promising route to measuring BAO over largevolumes at relatively low cost, but the technique is still under development. Photometricredshifts in optical imaging surveys can be used to measure BAO in the angular direction,though the typical distance precision is a factor of 3–4 lower compared to a well sampledspectroscopic survey of the same area, and angular BAO measurements do not directlyconstrain H(z). BAO distance measurements complement SN distance measurements byproviding absolute rather than relative distances (with precise calibration of rs from theCMB) and by having greater achievable precision at high redshift thanks to the increasingcomoving volume available. Theoretical modeling suggests that BAO measurementsfrom even the largest feasible redshift surveys will be limited by statistical rather thansystematic uncertainties.

Weak Gravitational Lensing: Gravitational light bending by a clustered distributionof matter shears the shapes of higher redshift background galaxies in a spatiallycoherent manner, producing a correlated pattern of apparent ellipticities. By studyingthe weak lensing signal for source galaxies binned by photometric redshift (estimatedfrom broad-band colors), one can probe the history of structure growth. “Cosmicshear” weak lensing uses the correlation of source ellipticities to deduce the clusteringof intervening matter. “Galaxy-galaxy lensing” (GGL) uses the correlation between ashear map and a foreground galaxy sample to measure the average mass profile aroundthe foreground galaxies, which can be combined with galaxy clustering to constraintotal matter clustering. For a specified expansion history, the predicted signals scaleapproximately as σ8Ω

αm, with α ≃ 0.3–0.5. The predicted signals also depend on the

distance-redshift relation, so weak lensing becomes more powerful in concert with SN orBAO measurements that can pin this relation down independently. The most challengingsystematics are shape measurement biases, biases in the distribution of photometricredshifts, and intrinsic alignments of galaxy orientations that could contaminate thelensing-induced signal. Weak lensing of CMB anisotropies is an increasingly powerfultool, in part because it circumvents many of these observational and astrophysicalsystematics. Predicting the large-scale weak lensing signal is straightforward in principle,but the number of independent modes on large scales is small, and the inferences are

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8 27. Dark energy

therefore dominated by sample variance. Exploiting small-scale measurements, for tighterconstraints, requires modeling the effects of complex physical processes such as starformation and feedback on the matter power spectrum. Strong gravitational lensing canalso provide constraints on dark energy, either through time delay measurements thatprobe the absolute distance scale, or through measurements of multiple-redshift lensesthat constrain distance ratios. The primary uncertainty for strong lensing constraints ismodeling the mass distribution of the lens systems.

Clusters of Galaxies: Like weak lensing, the abundance of massive dark-matter halosprobes structure growth by constraining σ8Ω

αm, where α ≃ 0.3–0.5. These halos can be

identified as dense concentrations of galaxies or through the signatures of hot (107–108K)gas in X-ray emission or SZ distortion of the CMB. The critical challenge in clustercosmology is calibrating the relation P (Mhalo|O) between the halo mass as predictedfrom theory and the observable O used for cluster identification. Measuring the stackedweak lensing signal from clusters has emerged as a promising approach to achievepercent-level accuracy in calibration of the mean relation, which is required for clusters toremain competitive with other growth probes. This method requires accurate modelingof completeness and contamination of cluster catalogs, projection effects on clusterselection and weak lensing measurements, and possible baryonic physics effects on themass distribution within clusters.

Redshift-Space Distortions (RSD) and the Alcock-Paczynksi (AP) Effect: Redshift-spacedistortions of galaxy clustering, induced by peculiar motions, probe structure growthby constraining the parameter combination f(z)σ8(z), where f(z) is the growth ratedefined by Eq. (27.3). Uncertainties in theoretical modeling of non-linear gravitationalevolution and the non-linear bias between the galaxy and matter distributions currentlylimit application of the method to large scales (comoving separations r >

∼ 10h−1Mpc orwavenumbers k <

∼ 0.2hMpc−1). A second source of anisotropy arises if one adopts thewrong cosmological metric to convert angles and redshifts into comoving separations,a phenomenon known as the Alcock-Paczynksi effect [25]. Demanding isotropy ofclustering at redshift z constrains the parameter combination H(z)DA(z). The mainchallenge for the AP method is correcting for the anisotropy induced by peculiar velocityRSD.

Low Redshift Measurement of H0: The value of H0 sets the current value of the criticaldensity ρc = 3H2

0/8πGN, and combination with CMB measurements provides a longlever arm for constraining the evolution of dark energy. The challenge in conventional H0

measurements is establishing distances to galaxies that are “in the Hubble flow,” i.e., farenough away that their peculiar velocities are small compared to the expansion velocityv = H0d. This can be done by building a ladder of distance indicators tied to stellarparallax on its lowest rung, or by using gravitational-lens time delays or geometricalmeasurements of maser data to circumvent this ladder.

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27. Dark energy 9

27.3.2. Dark Energy Experiments :

Most observational applications of these methods now take place in the context of largecosmological surveys, for which constraining dark energy and modified gravity theoriesis a central objective. Table 27.1 lists a selection of current and planned dark-energyexperiments, taken originally from the Snowmass 2013 Dark Energy Facilities review [26],which focused on projects in which the U.S. has either a leading role or significantparticipation. References and links to further information about these projects can befound in Ref. [26]. We have adjusted some of the dates in this Table relative to thosein Ref. [26] and added the European-led KiloDegree Survey (KiDS). Dates in the Tablecorrespond to the duration of survey observations, and the final cosmological resultsfrequently require 1–3 years of analysis and modeling beyond the end of data taking.

Table 27.1: A selection of major dark-energy experiments, based on Ref. [26].Abbreviations in the “Data” column refer to optical (Opt) or near-infrared (NIR)imaging (I) or spectroscopy (S). For spectroscopic experiments, the “Spec-z”column lists the primary redshift range for galaxies (gals), quasars (QSOs), or theLyman-α forest (LyαF). Abbreviations in the “Methods” column are weak lensing(WL), clusters (CL), supernovae (SN), baryon acoustic oscillations (BAO), andredshift-space distortions (RSD).

Project Dates Area/deg2 Data Spec-z Range Methods

BOSS 2008–2014 10,000 Opt-S 0.3–0.7 (gals) BAO/RSD

2–3.5 (LyαF)

KiDS 2011–2019 1500 Opt-I —— WL/CL

DES 2013–2019 5000 Opt-I —— WL/CL

SN/BAO

eBOSS 2014–2018 7500 Opt-S 0.6–2.0 (gal/QSO) BAO/RSD

2–3.5 (LyαF)

SuMIRE 2014–2024 1500 Opt-I WL/CL

Opt/NIR-S 0.8–2.4 (gals) BAO/RSD

HETDEX 2017–2023 450 Opt-S 1.9 < z < 3.5 (gals) BAO/RSD

DESI 2020–2025 14,000 Opt-S 0–1.7 (gals) BAO/RSD

2–3.5 (LyαF)

LSST 2022–2032 20,000 Opt-I —— WL/CL

SN/BAO

Euclid 2022–2028 15,000 Opt-I WL/CL

NIR-S 0.7–2.2 (gals) BAO/RSD

WFIRST 2025–2030 2200 NIR-I WL/CL/SN

NIR-S 1.0–3.0 (gals) BAO/RSD

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10 27. Dark energy

Beginning our discussion with imaging surveys, the Dark Energy Survey (DES)has observed 1/8 of the sky to a depth roughly 2 magnitudes deeper than the SloanDigital Sky Survey (SDSS), enabling weak lensing measurements with much greaterstatistical precision, cluster measurements calibrated by weak lensing, and angular BAOmeasurements based on photometric redshifts. With repeat imaging over a smaller area,DES has identified thousands of Type Ia SNe, which together with spectroscopic follow-updata enable significant improvements on the current state-of-the-art for supernova (SN)cosmology. Cosmological results from weak lensing and galaxy clustering analyses of thefirst year DES data are presented in Ref. [27] and discussed further below, while thefirst cosmological results from the DES supernova survey are presented in Ref. [28]. TheHyper-Suprime Camera (HSC) on the Subaru 8.2-m telescope is carrying out a similartype of optical imaging survey, probing a smaller area than DES but to greater depth.First cosmological results from HSC weak lensing are reported in Refs. [29,30]. TheHSC survey is one component of the Subaru Measurement of Images and Redshifts(SuMIRE) project. Beginning in the early 2020s, the dedicated Large Synoptic SurveyTelescope (LSST) will scan the southern sky to SDSS-like depth every four nights. LSSTimaging co-added over its decade-long primary survey will reach extraordinary depth,enabling weak lensing, cluster, and photometric BAO studies from billions of galaxies.Additionally, LSST time-domain monitoring will identify and measure light curves forthousands of Type Ia SNe per year.

Turning to spectroscopic surveys, the Baryon Oscillation Spectroscopic Survey (BOSS)and its successor eBOSS used fiber-fed optical spectrographs to map the redshift-spacedistributions of millions of galaxies and quasars. These 3-dimensional maps enable BAOand RSD measurements, and Lyman-α forest spectra of high-redshift quasars extendthese measurements to redshifts z > 2. As discussed below, the BOSS Collaboration hasnow published BAO and RSD analyses of its final data sets, and eBOSS has releasedBAO measurements from quasar clustering at z = 1–2. The Hobby-Eberly TelescopeDark Energy Experiment (HETDEX) uses integral field spectrographs to detect Lyman-αemission-line galaxies at z ≃ 1.9–3.5, probing a small sky area but a substantial comovingvolume. The Dark Energy Spectroscopic Instument (DESI) will follow a strategy similarto BOSS/eBOSS but on a much grander scale, using a larger telescope (4-m vs. 2.5-m)and a much higher fiber multiplex (5000 vs. 1000) to survey an order-of-magnitude moregalaxies. A new Prime Focus Spectrograph (PFS) for the Subaru telescope will enable thespectroscopic component of SuMIRE, with the large telescope aperture and wavelengthsensitivity that extends to the near-infrared (NIR) allowing it to probe a higher redshiftgalaxy population than DESI, over a smaller area of sky.

Compared to ground-based observations, space observations afford higher angularresolution and a far lower NIR sky background. The Euclid and WFIRST (Wide FieldInfrared Survey Telescope) missions will exploit these advantages, conducting large areaimaging surveys for weak lensing and cluster studies and slitless spectroscopic surveysof emission-line galaxies for BAO and RSD studies. WFIRST will also incorporate animaging and spectrophotometric supernova (SN) survey, extending to redshift z ≃ 1.7.Survey details are likely to evolve prior to launch, but in the current designs onecan roughly characterize the difference between the Euclid and WFIRST dark-energy

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27. Dark energy 11

experiments as “wide vs. deep,” with planned survey areas of 15,000 deg2 and 2200deg2, respectively. For weak lensing shape measurements, Euclid will use a single wideoptical filter, while WFIRST will use three NIR filters. The Euclid galaxy redshift surveywill cover a large volume at relatively low space density, while the WFIRST survey willprovide denser sampling of structure in a smaller volume. There are numerous synergiesamong the LSST, Euclid, and WFIRST dark energy programs, as discussed in Ref. [31].

27.4. Current Constraints on Expansion, Growth, and Dark

Energy

The last decade has seen dramatic progress in measurements of the cosmic expansionhistory and structure growth, leading to much tighter constraints on the parameters ofdark energy models. CMB data from the WMAP and Planck satellites and from higherresolution ground-based experiments have provided an exquisitely detailed picture ofstructure at the recombination epoch and the first CMB-based measures of low-redshiftstructure through lensing and SZ cluster counts. Cosmological supernova samples haveincreased in size from tens to many hundreds, with continuous coverage from z = 0 toz ≃ 1.4, alongside major improvements in data quality, analysis methods, and detailedunderstanding of local populations. BAO measurements have advanced from the firstdetections to 1–2% precision at multiple redshifts, with increasingly sophisticated methodsfor testing systematics, fitting models, and evaluating statistical errors. Advances inX-ray, SZ, and weak-lensing observations of large samples of galaxy clusters allow amulti-faceted approach to mass calibration, improving statistical precision but alsorevealing sources of astrophysical uncertainty. Cluster constraints have been joinedby the first precise matter-clustering constraints from cosmic-shear weak lensing andgalaxy-galaxy lensing, and by redshift-space distortion measurements that probe differentaspects of structure growth at somewhat lower precision. The precision of low-redshiftH0 measurements has sharpened from the roughly 10% error of the HST Key Project[32] to 2–4% in recent analyses.

As an illustration of current measurements of the cosmic expansion history, Fig. 27.1compares distance-redshift measurements from SN and BAO data to the predictions for aflat Universe with a cosmological constant. SN cosmology relies on compilation analysesthat try to bring data from different surveys probing distinct redshift ranges to a commonscale. Here we use the “joint light curve analysis” (JLA) sample of Ref. [35], who carriedout a careful intercalibration of the 3-year Supernova Legacy Survey (SNLS3, [36]) andthe full SDSS-II Supernova Survey [37] data in combination with several local supernovasamples and high-redshift supernovae from HST. Results from the Union2.1 sample [38],which partly overlaps JLA but has different analysis procedures, would be similar. Otherstate-of-the-art supernova data sets include the Pan-STARRS1 sample incorporated inthe PANTHEON compilation [39] and the first sample of spectroscopically confirmedsupernovae from DES [28]. For illustration purposes, we have binned the JLA datain redshift and plotted the diagonal elements of the covariance matrix as error bars,and we have converted the SN luminosity distances to an equivalent comoving angulardiameter distance. Because the peak luminosity of a fiducial SN Ia is an unknown freeparameter, the SN distance measurements could all be shifted up and down by a constant

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Figure 27.1: Distance-redshift relation measured from Type Ia SNe and BAOcompared to the predictions (black curve) of a flat ΛCDM model with Ωm = 0.315and h = 0.674, the best-fit parameters inferred from Planck CMB data [33]. Circlesshow binned luminosity distances from the JLA SN sample [35], multiplied by(1+z)−1 to convert to comoving angular diameter distance. Red squares show BAOdistance measurements from the 6dFGS, SDSS-II, BOSS, and eBOSS surveys (seetext for details and references). The lower panel plots residuals from the ΛCDMprediction, with dashed and dotted curves that show the effect of changing w by±0.1 while all other parameters are held fixed. Note that the SN data points canbe shifted up or down by a constant factor to account for freedom in the peakluminosity, while the BAO points are calibrated to 0.2% precision by the soundhorizon scale computed from Planck data. The errors on the BAO data pointsare approximately independent. In the upper panel, error bars are plotted only atz > 0.7 to avoid visual confusion.

multiplicative factor; cosmological information resides in the relative distances as afunction of redshift. The normalization used here corresponds to a Hubble parameterh = 0.674.

The z < 2 BAO data points come from the 6-degree-Field Galaxy Survey 6dFGSsurvey [40], the SDSS-II Main Galaxy Sample [41], the final galaxy clustering data set

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from BOSS [42], and the first BAO measurement from quasar clustering in eBOSS [43].For the 6dFGS, SDSS-II, and eBOSS data points, values of DV have been convertedto DA,c. The BOSS analysis measures DA,c directly; we have taken values from the“BAO only” column of table 7 of Ref. [42]. At z = 2.34 we plot DA,c measured fromthe BAO analysis of the eBOSS Lyman-α forest auto-correlation and cross-correlationwith quasars [44]. The BAO measurements are converted to absolute distances usingthe sound horizon scale rs = 147.09 Mpc from Planck 2018 CMB data, whose 0.18%uncertainty is small compared to the current BAO measurement errors. The BOSS galaxyand eBOSS Lyman-α forest analyses also measure H(z) at the same redshifts, providingfurther leverage on expansion history that is not captured in Fig. 27.1.

The plotted cosmological model has Ωm = 0.315 and h = 0.674, the best-fit valuesfrom Planck (TT+TE+EE+lowE+lensing) assuming w = −1 and Ωtot = 1 [33]. TheSN, BAO, and CMB data sets, probing a wide range of redshifts with radically differenttechniques, are for the most part mutually consistent with the predictions of a flatΛCDM cosmology. The eBOSS Lyman-α forest BAO measurements lie about 1.7σ fromthe Planck ΛCDM prediction [44], notably closer than the 2.3σ difference obtained withearlier BOSS data and discussed in the 2018 edition of this Review. Dotted and dashedcurves in the lower panel of Fig. 27.1 show the effect of changing w by ±0.1 with allother parameters held fixed, which leads to significantly worse agreement with the data.However, such a single-parameter comparison does not capture the impact of parameterdegeneracies or the ability of complementary data sets to break them, and if one insteadforced a match to CMB data by changing h and Ωm when changing w then the predictedBAO distances would diverge at z = 0 rather than converging there.

Figure 27.2, taken from Ref. [42], presents constraints on models that allow a freebut constant value of w with non-zero space curvature (owCDM, left panel) or the evolvingequation of state of Eq. (27.4) in a flat Universe (w0waCDM, right panel). Green contoursshow constraints from the combination of Planck 2015 CMB data and the JLA supernovasample. Gray contours show the combination of Planck with BAO measurements fromBOSS, 6dFGS, and SDSS-II. Red contours adopt a more aggressive analysis of the BOSSgalaxy data that uses the full shape (FS) of the redshift-space power spectrum andcorrelation function, modeled via perturbation theory, in addition to the measurement ofthe BAO scale itself. The full shape analysis improves the constraining power of the data,primarily because measurement of the Alcock-Paczynski effect on sub-BAO scales helpsto break the degeneracy between DA,c(z) and H(z). Blue contours show constraints fromthe full combination of CMB, BAO+FS, and SN data. Supernovae provide fine-grainedrelative distance measurements with good bin-by-bin precision at z < 0.7 (see Fig. 27.1),which is complementary to BAO for constraining redshift evolution of w. In both classesof model, the flat ΛCDM parameters (w = w0 = −1, ΩK = wa = 0) lie within the 68%confidence contour.

The precision on dark energy parameters depends, of course, on both the data beingconsidered and the flexibility of the model being assumed. For the owCDM model and thePlanck+BAO+FS+SN data combination, Ref. [42] finds w = −1.01 ± 0.04. Assuminga flat Universe and incorporating Planck 2018 data and DES Year 1 weak lensing, in

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Figure 27.2: Constraints on dark energy model parameters from combinationsof CMB, BAO, galaxy clustering, and supernova (SN) data, taken from Ref. [42].The left panel shows 68% and 95% confidence contours in the owCDM model, withconstant equation-of-state parameter w and non-zero space curvature ΩK ≡ 1−Ωtot.Green and gray contours show the combination of Planck CMB data with SN orBAO data, respectively. Red contours combine CMB, BAO, and the full shape (FS)of redshift-space galaxy clustering. Blue contours add SN data to this combination.The right panel shows confidence contours for the same data combinations in thew0waCDM model, which assumes a flat Universe and an evolving equation of statewith w(a) = w0 + wa(1− a).

addition to BAO and SN, Ref. [33] finds

w = −1.028± 0.031 . (27.5)

We consider either of these results to be a reasonable characterization of currentknowledge about the dark energy equation of state. In the w0waCDM model there is strongdegeneracy between w0 and wa, as one can see in Fig. 27.2. However, the value of wat the pivot redshift zp = 0.29 is well constrained by the Planck+BAO+FS+SN datacombination, with wp = −1.05± 0.06 [42]. The constraint on the evolution parameter,by contrast, remains poor even with this data combination, wa = −0.39 ± 0.34. Forexaminations of a wide range of dark energy, dark matter, neutrino content, and modifiedgravity models, see Refs. [33,42,46].

A flat ΛCDM model fit to Planck CMB data alone predicts H0 = 67.4 ±0.5 km s−1Mpc−1 (see Chapter 28 of this Review). This prediction and its error barare sensitive to the assumptions of constant dark energy and a flat Universe. However,by adding BAO and supernova data one can construct an “inverse distance ladder” tomeasure H0 precisely, even with a general dark energy model and free curvature [45].Ref. [47] applies this approach to obtain H0 = 67.8± 1.3 km s−1Mpc−1. As discussed

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in Sec. 24.3.1 of this Review, recent measurements from low-redshift data yield highervalues of H0. Figure 27.3 compares the CMB-anchored H0 estimates cited above todistance-ladder estimates that use Cepheid [48] or tip-of-the-red-giant-branch (TRGB) [49]stars to calibrate SNe Ia luminosities, and to an entirely independent estimate that usesgravitational-lens time delays [50]. The Cepheid and lensing estimates are discrepantwith the CMB-anchored estimates at a statistically significant level (Ref. [48] quotes 4.4σrelative to Planck ΛCDM), while the TRGB calibration yields an intermediate result thatis consistent with either the “high” or “low” values of H0.

The tension in H0 could reflect some combination of statistical flukes and systematicerrors in one or more of the data sets employed in these analyses. However, if theresolution lies in new physics rather than measurement errors, then this is probablyphysics that operates in the pre-recombination Universe, rescaling the BAO standardruler in a way that shifts the ΛCDM and inverse-distance-ladder values upward. Modelswith extra relativistic degrees of freedom or dark energy that is dynamically significant inthe early Universe can achieve this effect by increasing the early expansion rate, but theyare tightly constrained by the damping tail of CMB anisotropies. A finely tuned modelin which early dark energy decays rapidly after recombination can mitigate the tensionbetween CMB data and local H0 measurements [51], though it still prefers H0 valuesbelow those of Ref. [48].

The amplitude of CMB anisotropies is proportional to the amplitude of densityfluctuations present at recombination, and by assuming GR and a specified dark energymodel one can extrapolate the growth of structure forward to the present day to predictσ8. Probes of low-redshift structure yield constraints in the (σ8,Ωm) plane, which canbe summarized in terms of the parameter combination S8 ≡ σ8(Ωm/0.3)0.5. As discussedin earlier editions of this Review, many but not all weak-lensing and cluster studies todate yield S8 values lower than those predicted for Planck-normalized ΛCDM. The rightpanel of Fig. 27.3 illustrates the current state-of-play, comparing a selection of recentlypublished S8 estimates to the Planck+ΛCDM prediction of S8 = 0.832± 0.013.

The first four points show cosmic-shear weak-lensing estimates from the Deep LensSurvey [52], KiDS [53], DES [54], and HSC [29]. All of these estimates lie below thePlanck central value, though only the KiDS estimate is discrepant by ∼ 2σ. The nextfour points use galaxy-galaxy lensing in combination with galaxy clustering. Ref. [55]used weak-lensing data from SDSS imaging and the SDSS main galaxy redshift catalog,restricting the analysis to scales well described by perturbation theory. Refs. [56] and [57]used the same weak-lensing data but the BOSS LOWZ galaxy sample, and they employedtwo quite different approaches to model the clustering and lensing signals into the stronglynon-linear regime (r ≈ 1h−1Mpc) so that they could fully exploit the constrainingpower of the data. Ref. [58] found a strong discrepancy on these non-linear scalesbetween the predictions of a Planck-normalized ΛCDM model and the galaxy-galaxylensing of BOSS CMASS galaxies, measured from 250 deg2 of deep imaging from theCanada-France-Hawaii Telescope. Ref. [59], plotted in Figure 27.3, revisited these datawith a more general modeling approach and showed that the discrepancy persists over arange of redshift and galaxy stellar mass.

The third set of points in this panel shows S8 estimates that combine cosmic shear

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Figure 27.3: Tensions between low-redshift cosmological measurements and thepredictions of a CMB-normalized ΛCDM model. All error bars are 1σ; see text forobservational references. (Left) Open circles show values of H0 for flat ΛCDM withPlanck parameters or a general dark energy model constrained by a combination ofCMB, BAO, and supernova data. Filled circles show distance-ladder estimates basedon Cepheid or TRGB calibration or an independent estimate using gravitational-lenstime delays. (Right) Matter clustering characterized by the parameter combinationσ8(Ωm/0.3)0.5, as predicted by a Planck-normalized ΛCDM model (vertical dottedlines, black hexagon) and estimated from weak gravitational lensing using cosmicshear, galaxy-galaxy lensing and galaxy clustering, or a combination of the twoconstraints. Points of the same color are based on the same weak-lensing data. The“CMB lensing” point shows the value of σ8 for Ωm = 0.3 inferred from Planck CMBlensing, a measurement that is independent of the “Planck+ΛCDM” prediction andweighted to somewhat higher redshift than the other weak-lensing points.

with galaxy-galaxy lensing and galaxy clustering (a.k.a. “3 × 2” analyses because theycombine three 2-point correlations), restricted to fairly large scales in the perturbativeregime. Refs. [60] and [61] use KiDS weak-lensing data but two different galaxy samples;although they are statistically consistent with each other, the difference of their centralvalues illustrates the sensitivity to external data and analysis choices. Ref. [27] presentsconstraints from the 3 × 2 analysis of the Year 1 DES data, which yields an S8 valuelower than the Planck prediction but consistent at the ∼ 2σ level.

The “CMB lensing” point shows the matter-clustering amplitude inferred from PlanckCMB lensing; we have evaluated Eq. (38) of Ref. [62] at Ωm = 0.3 and adopted the samefractional error. It is important to emphasize that this is a measurement of low-redshiftclustering even though the background being lensed is the CMB. The CMB lensing kernelspeak at z <

∼ 1, with tails to z ∼ 5, so the effective redshift of the S8 measurement is

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somewhat higher than that of the other weak-lensing data points. The result is consistentwith the Planck+ΛCDM prediction at 1σ, and it is also consistent with the lower valuecorresponding to the mean or median of the optical weak-lensing measurements.

No one of these analyses provides convincing evidence of a conflict with the ΛCDMcosmological model. However, the case for such a conflict has grown stronger as thelow inferred clustering amplitude has persisted across multiple statistically independentweak-lensing surveys and multiple analysis methods. One possible explanation is thatseveral of the weak lensing surveys are affected by a common, unrecognized, systematicbias. Another possibility is a true deviation between the clustering growth extrapolatedforward from the early Universe and the clustering of matter at late times. Theconsistency of CMB lensing and Lyman-α forest measurements with ΛCDM clusteringpredictions suggests that any such deviation sets in mainly at z < 1, coinciding with theera of cosmic acceleration. Because the expansion history is well constrained by BAOand supernova data, it is difficult to change low-redshift matter clustering by simplychanging the equation of state of dark energy. Instead, a deviation between predictedand observed clustering might point towards modified gravity, decaying dark matter, orcoupling between dark matter and dark energy.

The next 2–3 years should see rapid progress on this conundrum. As the KiDS,DES, and HSC data sets grow in size, their statistical uncertainties will shrink, whichwill in turn enable more stringent internal cross-checks that test for consistent resultsfrom different redshift ranges, different scales, and different lens and source populations.Modeling methods that exploit non-linear scales will also be more stringently tested.Clusters of galaxies with weak-lensing mass calibration provide an alternative route to S8measurement, with competitive statistical precision. Recent cluster-based S8 estimatesspan a wide range, some of them consistent with Planck+ΛCDM and others implyinglower matter clustering, and we have not quoted results here because it is difficult to decidewhich are most reliable. However, the opportunity to combine multiple cluster sampleswith multiple weak-lensing surveys may lead to consistent and convincing measurementsin the near future. CMB lensing constraints will improve with higher angular resolutiondata from the South Pole Telescope and the Atacama Cosmology Telescope and theirsuccessors. Finally, the DESI survey will soon allow the first RSD-based measurementsof structure growth at the 1–2% level, providing an entirely distinct route to probe theclustering tension hinted at in Fig. 27.3.

27.5. Summary and Outlook

Figure 27.2 focuses on model parameter constraints, but to describe the observationalsituation it is more useful to characterize the precision, redshift range, and systematicuncertainties of the basic expansion and growth measurements. At present, supernovasurveys constrain distance ratios at the 1–2% level in redshift bins of width ∆z = 0.1 overthe range 0 < z < 0.6, with larger but still interesting error bars out to z ≃ 1.3. Thesemeasurements are currently limited by systematics tied to photometric calibration, dustreddening, host-galaxy correlations, and possible evolution of the SN population. BAOsurveys have measured the absolute distance scale (calibrated to the sound horizon rs)to 4% at z = 0.15, 1% at z = 0.38 and z = 0.61, and 2% at z = 2.3. Multiple studies

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have used clusters of galaxies or weak-lensing cosmic shear or galaxy-galaxy lensing tomeasure a parameter combination σ8Ω

αm with α ≃ 0.3–0.5. The estimated errors of the

most recent studies, including both statistical contributions and identified systematicuncertainties, are 3–5%. RSD measurements constrain the combination f(z)σ8(z), andrecent determinations span the redshift range 0 < z < 0.9 with typical estimated errorsof about 10%. These errors are dominated by statistics, but shrinking them further willrequire improvements in modeling non-linear effects on small scales. Distance-ladderestimates of H0 now span a small range, using overlapping data but distinct treatmentsof key steps; individual studies quote uncertainties of 2–5%, with similar statistical andsystematic contributions. Planck data and higher resolution ground-based experimentsnow measure CMB anisotropies with exquisite precision; for example, CMB measurementsnow constrain the physical size of the BAO sound horizon to 0.2% and the angular scaleof the sound horizon to 0.01%.

A flat ΛCDM model with standard radiation and neutrino content can fit the CMBdata and the BAO and SN distance measurements to within their estimated uncertainties.The CMB+BAO parameters for this model are in significant tension with some butnot all recent measurements of H0 determined from low-redshift data. The discrepancycould reflect underestimated systematic errors in one or more of the input data sets. Ifthe conflict is real, then it may point to new physics in the pre-recombination Universethat rescales the sound horizon, such as early dark energy or extra relativistic degreesof freedom. Many measurements of low-redshift matter clustering from weak lensing liebelow the predictions of a ΛCDM model extrapolated forward from the Planck CMBanisotropies. No one analysis presents a convincing conflict, but the difference persistsacross several independent data sets and analysis methods. If real, this discrepancycould point towards modified gravity, decaying dark matter, or coupling between darkmatter and dark energy. However, none of the tensions present in the data yet providescompelling evidence for new physics.

Analyses of the final KiDS and DES weak-lensing data sets and the expanding HSCweak-lensing data set should yield measurements of matter clustering that have sharperstatistical precision and more stringent tests of internal consistency. Fully exploiting thesedata will require further development of accurate models of matter clustering, galaxyclustering, and weak lensing by galaxy clusters into the fully non-linear regime, includingrobust methods of accounting for uncertainties in the baryonic mass distribution. It willalso require further progress on the thorny challenge of photometric redshift calibrationso that these uncertainties do not dominate the error budget. Higher signal-to-noiseCMB lensing maps cross-correlated with galaxies will provide independent tests thatavoid some of the systematic uncertainties of optical weak lensing. H0 measurementswill improve with increasing numbers of Cepheid or TRGB distances to supernovahost galaxies, improving Gaia parallaxes of Galactic Cepheids, increasing numbersof strong gravitational-lens time delays, and continued attention to the systematicuncertainties in each method. Improving measurements of the CMB damping tail fromground-based experiments will provide increasingly strong constraints on solutionsinvolving pre-recombination physics.

After beginning operations in early 2020, the DESI galaxy redshift survey will quickly

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exceed the size of the existing SDSS and BOSS surveys, enabling high precision BAOmeasurements of expansion history at z ≈ 0.7–1.4 and, for the first time, percent-levelmeasurements of structure growth through RSD. Precise BAO and RSD measurementsat higher redshifts will come from DESI Lyman-α forest maps and the HETDEX andSubaru PFS galaxy surveys. The BAO measurements will complement increasinglyprecise measurements of the relative distance scale at z < 1 from the DES photometricsupernova sample and from improved local supernova samples (z < 0.1) that provide alow-redshift anchor. Large galaxy samples will also enable more powerful applications ofthe Alcock-Paczynski effect, which can amplify the power of BAO and supernova distancemeasurements by converting them to constraints on the expansion rate H(z).

The early-to-mid 2020s will see another major leap in observational capabilities withthe advent of LSST, Euclid, and WFIRST. LSST will be the ultimate ground-basedoptical weak-lensing experiment, measuring several billion galaxy shapes over 20,000 deg2

of the southern hemisphere sky, and it will detect and monitor many thousands of SNeper year. Euclid and WFIRST also have weak lensing as a primary science goal, takingadvantage of the high angular resolution and extremely stable image quality achievablefrom space. Both missions plan large spectroscopic galaxy surveys, which will providebetter sampling at high redshifts than DESI or PFS because of the lower infrared skybackground above the atmosphere. WFIRST is also designed to carry out what shouldbe the ultimate supernova cosmology experiment, with deep, high resolution, near-IRobservations and the stable calibration achievable with a space platform. The 2020s willalso see dramatic advances in CMB lensing from the Simons Observatory and, potentially,CMB-S4 and/or a space-based probe; cross-correlation with galaxy surveys allows precisetomographic measurements of clustering as a function of redshift.

If the anomalies suggested in Fig. 27.3 are real, then the experiments of the 2020swill map out their redshift, scale, and environment dependence in great detail, providingdetailed empirical constraints on dynamical dark energy or modified gravity models. Ifthese tensions dissipate with improved measurements, then the experiments of the 2020swill achieve much more stringent tests of the ΛCDM paradigm, with the potential toreveal deviations that are within the statistical uncertainties of current data. The criticalclue to the origin of cosmic acceleration could also come from a surprising direction, suchas laboratory or solar-system tests that challenge GR, time variation of fundamental“constants,” or anomalous behavior of gravity in some astronomical environments.Experimental advances along these multiple axes could confirm today’s relatively simple,but frustratingly incomplete, “standard model” of cosmology, or they could force yetanother radical revision in our understanding of energy, or gravity, or the spacetimestructure of the Universe.

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